| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Standard +0.3 This is a straightforward application of binomial distribution with standard calculations. Parts (a) and (b) are routine binomial probability computations using n=20, p=0.3. Part (c) adds one layer by treating 'bags with >3 faulty bolts' as a new binomial trial, but this is a standard S2 technique requiring no novel insight—just careful application of the binomial formula twice. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Let \(X\) be the number of faulty bolts. \(P(X \leq 2) - P(X \leq 1) = 0.0355 - 0.0076 = 0.0279\), or \((0.3)^2(0.7)^{18}\frac{20!}{18!2!} = 0.0278\) | M1, A1 |
| (b) | \(1 - P(X \leq 3) = 1 - 0.1071 = 0.8929\) | M1, A1 |
| (c) | \(\frac{10!}{4!6!}(0.8929)^6(0.1071)^4 = 0.0140\) | M1, A1\(\checkmark\), A1 |
## Question 2:
**(a)** | Let $X$ be the number of faulty bolts. $P(X \leq 2) - P(X \leq 1) = 0.0355 - 0.0076 = 0.0279$, or $(0.3)^2(0.7)^{18}\frac{20!}{18!2!} = 0.0278$ | M1, A1 | M1: either attempting $P(X\leq2)-P(X\leq1)$, or attempting binomial with $(p)^2(1-p)^{18}\frac{20!}{18!2!}$ with a value of $p$. A1: awrt 0.0278 or 0.0279.
**(b)** | $1 - P(X \leq 3) = 1 - 0.1071 = 0.8929$ | M1, A1 | M1: attempting $1-P(X\leq3)$. A1: awrt 0.893.
**(c)** | $\frac{10!}{4!6!}(0.8929)^6(0.1071)^4 = 0.0140$ | M1, A1$\checkmark$, A1 | M1: for $k(p)^6(1-p)^4$; A1$\checkmark$: $\frac{10!}{4!6!}(\text{their part b})^6(1-\text{their part b})^4$; A1: awrt 0.014. | (3)
---2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are
\begin{enumerate}[label=(\alph*)]
\item exactly 2 faulty bolts,
\item more than 3 faulty bolts.
These bolts are sold in bags of 20. John buys 10 bags.
\item Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2008 Q2 [7]}}